Finite Group Databasebeta

← #10696760

Subgroups — #10696760

Order 768.

544 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3215C211yesmaximal, characteristic, complemented
2#83361C211yesmaximal, characteristic, complemented
3#82943C211yesmaximal, characteristic
4#71825C211yesmaximal, characteristic, complemented
5#71720C211yesmaximal, characteristic, complemented
6#70801C211yesmaximal, characteristic, complemented
7#67753C211yesmaximal, characteristic, complemented
8#10121—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C2×#942—21no—
12C2×#932—21no—
13#5477C2211yescharacteristic
14C3×#497—21no—
15C3×#489—21no—
16C3×#464—22no—
17C3×#451—21no—
18C3×#450—21no—
19C3×#432C2211yescharacteristic
20#5087—21no—
21#5084—21no—
22#5067—21no—
23#5064—21no—
24#5060—21no—
25#5059—22no—
26#5058—21no—
27#5003—22no—
28#4926C2211yescharacteristic
29#4925C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4535—22no—
32#4525—22no—
33#4483—22no—
34#3235—31no—
35#3215D311yescharacteristic, complemented
36#3153—31no—
37#1496—31no—
38#1485—31no—
39#1469—31no—
40#1291—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C2×#235—21no—
44C2×C3×Q16—22no—
45C2×C3×SD16—22no—
46C3×C4⋊Q8D411yescharacteristic
47C3×C4⋊Q8D411yescharacteristic
48C3×#125—21no—
49C3×#124—21no—
50C3×#124—21no—
51C3×#124—21no—
52C3×#124—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×Q8—21no—
57C3×#117—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C42C2311yescharacteristic
61C2×#216—22no—
62#942—44no—
63C2×#215—22no—
64C2×#214—22no—
65#932—44no—
66C2×#213—22no—
67#930—21no—
68#930—21no—
69C2×#208—21no—
70#875—21no—
71#874D411yescharacteristic
72C4×D12—21no—
73C4×D12—21no—
74#869D411yescharacteristic
75C3×C4⋊C8D412yes—
76C3×#103—22no—
77C3×#102—22no—
78C3×C22⋊C8—21no—
79C3×#97—21no—
80#830—21no—
81#830D412yes—
82#809—22no—
83#809—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89#803—21no—
90C4×C3⋊C8—21no—
91C4×C3⋊C8D412yes—
92C2×#137—61no—
93C2×AGL(1,ℤ/8)—61no—
94#533C2×D311yescharacteristic
95#532—31no—
96C2×#124C2×D311yescharacteristic
97#501—61no—
98#497—21no—
99#497—61no—
100#496—61no—
101#495—62no—
102#490—61no—
103#489—21no—
104#482—62no—
105#476—61no—
106#474—61no—
107#464—22no—
108#451—21no—
109#450—21no—
110#432—31no—
111#432C2×D311yescharacteristic
112#431—31no—
113C4×M16—31no—
114Q8⋊C8—62no—
115D4⋊C8—62no—
116#323—62no—
117C24×C3—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—42no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—42no—
126C22×C3×C4—21no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C2×#67—41no—
131#235—42no—
132#235—42no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×C4×D3—41no—
137C2×Dic6—21no—
138C3×Q16—82no—
139C3×SD16—82no—
140C2×C3×C8—22no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C42—41no—
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157#216—44no—
158#215—44no—
159#214—44no—
160#213—44no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165#208—24no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—41no—
169C2×#41—61no—
170C22×Q8#6711yescharacteristic
171C22×D4#6711yescharacteristic
172#137—124no—
173AGL(1,ℤ/8)—124no—
174C2×Q16—22no—
175C2×Q16—62no—
176C2×SD16—22no—
177C2×SD16—62no—
178C2×SD16—62no—
179C2×D8—62no—
180C2×M16—61no—
181C4⋊Q8—31no—
182C4⋊Q8#6711yescharacteristic
183C4⋊Q8#6711yescharacteristic
184#127—31no—
185#125—21no—
186#124—21no—
187#124—21no—
188#124—21no—
189#124—21no—
190#124—61no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193#121—61no—
194#121—61no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×D4—61no—
198C4×D4—61no—
199#117—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C42C22×D311yescharacteristic
203#107—62no—
204#106—62no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8#6712yes—
208#103—22no—
209#102—22no—
210#102—62no—
211#102—62no—
212C22⋊C8—21no—
213C22⋊C8—32no—
214C22⋊C8—61no—
215#97—21no—
216#97—61no—
217#97—61no—
218C4×C8—32no—
219C4×C8—61no—
220C23×C3—42no—
221C23×C3—41no—
222C23×C3—81no—
223C23×C3—81no—
224C23×C3#12111yescharacteristic
225C22×D3—41no—
226C3×Q8—42no—
227C3×Q8—42no—
228C3×Q8—42no—
229C3×Q8—42no—
230C3×D4—42no—
231C3×D4—42no—
232C2×C3×C4—22no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—81no—
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12711yescharacteristic
245#67—82no—
246C2×Dic3—41no—
247D12—41no—
248D12—41no—
249D12—42no—
250D12—42no—
251C4×D3—82no—
252Dic6—41no—
253Dic6—41no—
254C3×C8—42no—
255C3⋊C8—24no—
256C3⋊C8—42no—
257C3⋊C8—81no—
258C24—21no—
259#41—122no—
260#41—122no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—42no—
266C2×Q8—61no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—42no—
270C2×D4—61no—
271C2×D4—61no—
272C2×D4—61no—
273C2×D4—121no—
274C22×C4—21no—
275C22×C4—121no—
276C22×C4D3×D411yescharacteristic
277C22×C4#23711yescharacteristic
278C22×C4C2×#6711yescharacteristic
279Q16—82no—
280Q16—124no—
281SD16—82no—
282SD16—124no—
283SD16—124no—
284D8—124no—
285M16—64no—
286C2×C8—22no—
287C2×C8—62no—
288C2×C8—62no—
289C2×C8—121no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C42—41no—
306C42D3×D411yescharacteristic
307C42D3×D411yescharacteristic
308C42C2×#6711yescharacteristic
309C42C2×#6711yescharacteristic
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3—162no—
317C22×C3#39211yescharacteristic
318C2×D3—81no—
319C2×D3—81no—
320C2×D3—81no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—81no—
327C3×C4—81no—
328C3×C4—81no—
329C3×C4#45312yes—
330Dic3—81no—
331C23—42no—
332C23—41no—
333C23—81no—
334C23—81no—
335C23—121no—
336C23#93911yescharacteristic
337Q8—42no—
338Q8—42no—
339Q8—42no—
340Q8—42no—
341Q8—121no—
342Q8—121no—
343D4—42no—
344D4—42no—
345D4—121no—
346D4—121no—
347D4—122no—
348D4—122no—
349D4—242no—
350C2×C4—22no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—81no—
357C2×C4—121no—
358C2×C4—242no—
359C2×C4#93711yescharacteristic
360C2×C4#93711yescharacteristic
361C2×C4#93811yescharacteristic
362C2×C4#93911yescharacteristic
363C2×C4#94011yescharacteristic
364C2×C4#95311yescharacteristic
365C8—42no—
366C8—64no—
367C8—122no—
368C8—241no—
369C2×C3—41no—
370C2×C3—161no—
371C2×C3#200211yescharacteristic
372C2×C3#200911yescharacteristic
373C2×C3#201711yescharacteristic
374D3—161no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—162no—
382C22—241no—
383C22—241no—
384C22—241no—
385C22#522311yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—241no—
395C4#524112yes—
396C3#1012111yescharacteristic, complemented, Sylow
397C2—41no—
398C2—161no—
399C2—481no—
400C2#7185611yescharacteristic
401C2#7211111yescharacteristic
402C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3215C2
socleC22×C3#392
prime radical#82943C2
hypercenter#3215D3
hypocenterC3#10121
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3215

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#3215C2semidirect
2. #83361C2semidirect
4. #71825C2semidirect
5. #71720C2semidirect
6. #70801C2semidirect
7. #67753C2semidirect
35. #3215D3semidirect
396. C3#10121semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#32152
#833612
#829432
#718252
#717202
#708012
#677532
#101213